At one or two points in the preceding chapters, we have referred to the existence or non-existence of algorithms to solve particular problems. It's easy to see how one proves that an algorithm exists: one simply constructs it, and demonstrates that it does the job one wants it to do. But how can one prove that no algorithm exists for a particular problem?

Clearly, in order to do this, we are going to have to be more precise than hitherto about what we mean by an algorithm. Informally, we can think of an algorithm as some calculation which a computer could be programmed to carry out; but, in order to make this precise, we need a precise definition of what we mean by a computer. In fact, our ‘idealized’ mathematical model of a computer will be a pretty feeble thing in comparison with most physically existing computers (we are not, on this theoretical level, interested in questions of speed or efficiency of computation, and so for simplicity we shall give our computer only the minimum of features needed for it to function at all); but in one respect it will be more powerful than the largest computer ever built – it will be able to handle arbitrarily large natural numbers.